On the surface of the paraboloid, we consider a family of orthogonal polynomials with respect to the
weight function \(\varpi (t) = t^{\beta }(1-t)^{\gamma }\) , which shall be called the Jacobi polynomials on the paraboloid.
These polynomials will be shown to be eigenfunctions of a second order differential operator but with the
eigenvalues depending on two indices, the degree of the polynomials as well as another index that
depends on the particular choice of the orthogonal basis, in contrast to the cone and the hyperboloid for
which the corresponding eigenvalue property is independent of the choice of orthogonal bases. The
Jacobi polynomials on the paraboloid also do not possess an explicit addition formula. What they do
have is a connection to an orthogonal structure on the parabolic domain
\({\mathbb {U}}= \lbrace (x_1,x_2): x_1^2 \le x_2 \le 1\rbrace \)  in \({\mathbb {R}}^2\) , bounded by the parabola \(x_2 =x_1^2\) 
and the line \(x_2 =1\) . A family of orthogonal polynomials on \({\mathbb {U}}\) , called the Jacobi polynomials on
the parabolic domain, was first considered in [1]} and they satisfy a product formula as shown
in [2]}. The latter was used to study the Fourier orthogonal series in [3]}, where the
essential ingredient is an addition formula that holds when one argument of the reproducing kernel is
at the corner \((1,1)\)  of the domain. Our essential realization is that the reproducing kernel of the
orthogonal polynomials on the paraboloid can be expressed in terms of the reproducing kernel on the
parabola domain, which provides us with the tool for studying the Fourier orthogonal series on the
paraboloid. In particular, it allows us to study the convergence of the Cesàro means of the series.
We shall also show that the connection to the structure on \({\mathbb {U}}\)  also extends to the solid paraboloid,
which allows us to carry out our study of the Fourier orthogonal series on the solid paraboloid.
